91Ó°ÊÓ

A quadratic equation is a fundamental concept in algebra, and is expressed in a standard form:

• \[ ax^2 + bx + c = 0 \]

Here, \(a\), \(b\), and \(c\) are known as the coefficients of the equation. The term \(ax^2\) represents the quadratic term, \(bx\) the linear term, and \(c\) the constant term.
The presence of the squared term \(ax^2\) makes it a quadratic, distinguishing it from linear equations which do not contain squared terms. Solving a quadratic typically involves finding the values of \(x\) that satisfy the equation — these values are often referred to as the 'roots' of the equation.
Quadratic equations can be solved by different methods, but one of the most systematic and widely used methods is the quadratic formula. This formula provides an exact solution for the roots, given that the equation is set to zero.

The roots of a quadratic equation are essentially the solutions to the equation, and these can be found using the quadratic formula:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our specific exercise, we are working with this formula to find the roots, but focusing on only one specific root expression:

• \[ \frac{-b - \sqrt{b^2 - 4ac}}{2a} \]

The part inside the square root, known as the discriminant \((b^2 - 4ac)\), is crucial. It determines the nature of the roots. If this discriminant is:

• Positive: there are two distinct real roots.
• Zero: there is one real root (or a double root).
• Negative: the equation has two complex roots.

For our exercise, the process of solving involves substituting the values of \(a = 2\), \(b = 9\), and \(c = -5\) into the formula and calculating the expression step-by-step to uncover one of these roots.

Simplifying expressions is an essential step when working with complex algebraic equations like quadratic equations. It involves breaking down the expression into a more manageable form.
In the given exercise, we were required to simplify the expression resulting from the insertion of specific values into the quadratic formula.### Key Steps to Simplification:1. **Substitute Known Values:** Plug the given values of \(a=2\), \(b=9\), and \(c=-5\) into the base expression \(\frac{-b - \sqrt{b^{2}-4ac}}{2a}\). 2. **Simplify Under the Square Root:** Calculate \(b^2 = 81\) and \(4ac = -40\). Then add these to update the inside of the square root to \(81 + 40 = 121\). 3. **Evaluate the Square Root:** Solve for \(\sqrt{121} = 11\) simplifying the expression to \(\frac{-9 - 11}{4}\). 4. **Finalize Simplification:** Complete the arithmetic in the fraction to find \(\frac{-20}{4} = -5\).Throughout this process, simplification is vital as it leads us to the final, clean result that represents one of the roots of our quadratic equation. This step-by-step simplification ensures that errors are minimized and the algebraic expression is easily interpretable.